Theorem f1omo 49368

Description: There is at most one element in the function value of a constant function whose output is 1_o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 49367 assuming ax-un 7689 (see f1omoALT 49370). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)

Hypothesis

Ref	Expression
f1omo.1	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omo	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo

Step	Hyp	Ref	Expression
1		1oex 8415	. . . 4 ⊢ 1o ∈ V
2		eqid 2736	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 49366	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 49289	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		df1o2 8412	. . . . . 6 ⊢ 1o = {∅}
6	5	eqeq2i 2749	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o ↔ ((𝐴 × {1o})‘𝑋) = {∅})
7		mosn 49288	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = {∅} → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
8	6, 7	sylbi 217	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
9	4, 8	jaoi 858	. . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
10	3, 9	ax-mp 5	. 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
11		f1omo.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
12	11	fveq1d 6842	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
13	12	eleq2d 2822	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
14	13	mobidv 2549	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
15	10, 14	mpbiri 258	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  ∅c0 4273  {csn 4567   × cxp 5629  ‘cfv 6498  1_oc1o 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-1o 8405

This theorem is referenced by:  indthinc  49937  prsthinc  49939